Answer:
p = 5/7
Step-by-step explanation:
The given function is:
[tex]g(x) = -3x^{2} - 2x + 8[/tex] for -4 ≦ x < 1
[tex]g(x) = -2x + 7p[/tex] for 1 ≦ x ≦ 5
Part a)
A continuous function has no breaks, jumps or holes in it. So, in order for g(x) to be continuous, the point where g(x) stops during the first interval -4 ≦ x < 1 must be equal to the point where g(x) starts in the second interval 1 ≦ x ≦ 5
The point where, g(x) stops during the first interval is at x = 1, which will be:
[tex]-3(1)^{2}-2(1)+8=3[/tex]
The point where g(x) starts during the second interval is:
[tex]-2(1)+7(p) = 7p - 2[/tex]
For the function to be continuous, these two points must be equal. Setting them equal, we get:
3 = 7p - 2
3 + 2 = 7p
p = [tex]\frac{5}{7}[/tex]
Thus the value of p for which g(x) will be continuous is [tex]\frac{5}{7}[/tex].
Part b)
We have to find p by setting the two pieces equal to each other. So, we get the equation as:
[tex]-3x^{2}-2x+8=-2x+7p\\\\ -3x^{2}+8=7p[/tex]
Substituting the point identified in part (a) i.e. x=1, we get:
[tex]-3(1)^{2}+8=7p\\\\ 5=7p\\\\ p=\frac{5}{7}[/tex]
This value agrees with the answer found in previous part.
